Then that alone should be enough to convince you that there is more to consider here than just inputting a string in to SHA256 and being happy that whatever it outputs is secure enough to use as your entropy source.
I refuse to accept that a random number once used as input in SHA256 gives non-cryptographically-secure result, not because I put myself above experts, but because experts say it. Take an ECDSA signature. In most Bitcoin wallets, value k is no longer generated using an RNG. Instead, it's a hash of the private key and the message. Quoting the important part from RFC 6979 (which is the standard most such software follow):
Quote from: https://www.rfc-editor.org/rfc/rfc6979
This document defines a deterministic digital signature generation procedure. Such signatures are compatible with standard Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA) digital signatures and can be processed with unmodified verifiers, which need not be aware of the procedure described therein. Deterministic signatures retain the cryptographic security features associated with digital signatures but can be more easily implemented in various environments, since they do not need access to a source of high-quality randomness.
Besides ECDSA signatures, the second half part of HD wallets is deterministic. Hence, the entire structure relies on the fact that random numbers passed through hash functions are cryptographically secure.
But damn it Jim!
Is this in a manner of speaking? Who's Jim? 